We develop a theory for solving continuous time optimal stopping problems for non-linear expectations. Our motivation is to consider problems in which the stopper uses risk measures to evaluate future rewards.
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48 results for “expected stopping time”
This work bounds the run-time of nonconvex optimization with early stopping.
problem Bounding the expected run-time of nonconvex optimization with early stopping.
method Derives conditions for well-defined early stopping based on validation function norms and bounds the expected number of iterations and gradient evaluations.
result Guarantees the validity of early stopping and provides bounds on the expected run-time for various optimization algorithms.
Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEsmath.PR
We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) -expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respect…
We study the existence of optimal actions in a zero-sum game between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem for a class of sublinear expectations such as the -expectation. We show that …
Analyzes Lévy flights on manifolds for finding small targets.
problem Finding small targets using Lévy flights on various manifolds.
method Analytic description of Lévy flights on closed Riemannian manifolds, including asymptotics of expected stopping time.
result Computes the expected time for finding a small target by Lévy flight on surfaces.
A framework for robust exploration in reinforcement learning under ambiguity.
problem Optimal stopping under ambiguity in reinforcement learning.
method Continuous-time robust reinforcement learning framework using -expectation and backward stochastic differential equations.
result Constructs a robust exploratory stopping time approximating the optimal stopping time under ambiguity.
We study optimal double stopping problems driven by a Brownian bridge. The objective is to maximize the expected spread between the payoffs achieved at the two stopping times. We study several cases where the solutions can be solved explicitly by strategies of threshold type.
We analyze an optimal stopping problem with random maturity under a nonlinear expectation with respect to a weakly compact set of mutually singular probabilities . The maturity is specified as the hitting time to level of some continuous index process at which the payoff process is even allowed to have…
Study optimal stopping times under regime-switching models with constraints.
problem Optimal stopping times for discounted payoffs on a regime-switching geometric Brownian motion.
method Solve variational inequality to find value functions and optimal thresholds.
result Existence and expressions of optimal stopping times under specific conditions.
Stop-loss and Leverage in optimal Statistical Arbitrage with an application to Energy marketq-fin.PM
In this paper we develop a statistical arbitrage trading strategy with two key elements in hi-frequency trading: stop-loss and leverage. We consider, as in Bertram (2009), a mean-reverting process for the security price with proportional transaction costs; we show how to introduce stop-loss and leverage in an optimal t…
Continuous-time optimal stopping solved with deep reinforcement learning
problem Optimal stopping problems in continuous time
method CARLOS (Continuous-time Adaptive Reinforcement Learning for Optimal Stopping)
result Higher prices than existing Bermudan solvers, approaching American upper bound
New algorithms improve stopping time for best arm identification.
problem Efficiently identifying the best alternative in experiments.
method Proposed algorithms with exponential-tailed stopping time.
result Proved that some algorithms never stop, leading to new methods.
This paper focuses on numéraire portfolio and log-optimal portfolio (portfolio with finite expected utility that maximizes the expected logarithm utility from terminal wealth), when a market model -specified by its assets' price and its flow of information - is stopped at a random time $τ…
Inspired by recent work of P.-L. Lions on conditional optimal control, we introduce a problem of optimal stopping under bounded rationality: the objective is the expected payoff at the time of stopping, conditioned on another event. For instance, an agent may care only about states where she is still alive at the time …
We introduce a simple stochastic volatility model, whose novelty consists in taking into account hitting times of the asset price, and study the optimal stopping problem corresponding to a put option whose time horizon (after the asset price hits a certain level) is exponentially distributed. We obtain explicit optimal…
In this paper we propose and solve an optimal dividend problem with capital injections over a finite time horizon. The surplus dynamics obeys a linearly controlled drifted Brownian motion that is reflected at the origin, dividends give rise to time-dependent instantaneous marginal profits, whereas capital injections ar…
A new stopping criterion for active learning based on deterministic generalization bounds.
problem Determining the optimal stopping point for active learning when data acquisition is costly.
method The proposed stopping criterion is based on the difference in expected generalization errors and hypothesis testing, derived from PAC-Bayesian theory.
result The proposed stopping criterion effectively stops active learning by combining an upper bound with a statistical test.
A new method uses deep learning for optimal stopping problems.
problem Solving optimal stopping problems in financial mathematics.
method Deep primal-dual BSDE framework with a novel loss function.
result The method provides a true upper bound for the optimal value.
The paper analyzes early stopping in linear regression and shows it's equivalent to ridge regularization.
problem Understanding the effect of early stopping on linear regression models.
method Characterization of gradient descent dynamics and analysis of excess risk.
result Early stopped solution is equivalent to minimum norm solution for a generalized ridge regularized problem.
The value of foresightmath.PR
Suppose you have one unit of stock, currently worth 1, which you must sell before time . The Optional Sampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value we get when we sell will be 1. Suppose however that we are able to see units of time into the future, and ba…
Optimal retirement timing and consumption under shortfall risk management
problem Optimal portfolio, consumption, and endogenous early retirement problem
method Maximizing expected lifetime consumption utility while managing the maximum wealth shortfall relative to a benchmark
result Geometric structure of the stopping set and feedback-form optimal retirement boundary
Proposes data-driven methods for estimating conditional expectations.
problem Estimating conditional expectations when underlying density is unknown.
method Data-driven techniques to directly estimate conditional expectations from training data.
result Extends data-driven method to solve nonlinear equations in stochastic optimization.
New method solves optimal stopping problems using rough path signatures.
problem Optimal stopping problems in finance and other fields.
method Using rough path signatures and deep neural networks.
result Solves optimal stopping problems efficiently under minimal assumptions.
We study a robust optimal stopping problem with respect to a set $\cP$ of mutually singular probabilities. This can be interpreted as a zero-sum controller-stopper game in which the stopper is trying to maximize its pay-off while an adverse player wants to minimize this payoff by choosing an evaluation criteria from $\…
It is known that the decision to purchase an annuity may be associated to an optimal stopping problem. However, little is known about optimal strategies, if the mortality force is a generic function of time and if the `subjective' life expectancy of the investor differs from the `objective' one adopted by insurance com…
An unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account how ambiguity-averse an agent is. This inclusion of ambiguity attitude, via an -maxmin nonlinear expectation, renders the stopping problem time-inconsistent. We look for subgame perfect…
Study examines how slight model changes affect multi-period optimization outcomes.
problem Effect of small probabilistic model changes on multi-period optimization problems.
method Adapted Wasserstein distance for measuring changes, explicit first-order approximations proved.
result Explicit first-order approximations for multi-period stochastic optimization and optimal stopping problems.
Study best arm identification with limited precision sampling in bandits.
problem Limited precision sampling in multi-armed bandit problems.
method Proposed a modified tracking-based algorithm to handle non-unique optimal allocations and presented non-asymptotic bounds.
result Asymptotically optimal tracking-based algorithm for best arm identification.
Stop-loss rules are often studied in the financial literature, but the stop-loss levels are seldom constructed systematically. In many papers, and indeed in practice as well, the level of the stops is too often set arbitrarily. Guided by the overarching goal in finance to maximize expected returns given available infor…
Investment strategy optimization from discrete to continuous models.
problem Optimizing investment strategies and stopping times in both continuous and discrete settings.
method Characterized value functions via quadratic reflected BSDEs for continuous case, discretized BSDEs for discrete case, and derived uniform convergence rates.
result Uniform convergence and rate from discrete to continuous quadratic reflected BSDEs.
Optimal reinsurance strategy with fixed cost and exponential preferences.
problem Maximizing expected utility of terminal wealth with fixed reinsurance cost.
method Two-step procedure: stochastic control and optimal stopping problem.
result Deterministic optimal strategy depends on model parameters.
Optimizes quickest detection of drift in Brownian motion with false negatives.
problem Quickest detection of drift in Brownian motion with false negatives.
method Formulated as an optimal multiple stopping problem, then equivalent to a recursive optimal stopping problem, solved using free boundary methods.
result Explicit formulae for expected cost and optimal strategy found.
We study the optimal stopping problem of pricing an American Put option on a Zero Coupon Bond (ZCB) in the Musiela's parametrization of the Heath-Jarrow-Morton (HJM) model for forward interest rates. First we show regularity properties of the price function by probabilistic methods. Then we find an infinite dimensional…
This paper proposes and analyses a new multilevel Monte Carlo method for the estimation of mean exit times for multi-dimensional Brownian diffusions, and associated functionals which correspond to solutions to high-dimensional parabolic PDEs through the Feynman-Kac formula. In particular, it is proved that the complexi…
A reinsurance contract should address the conflicting interests of the insurer and reinsurer. Most of existing optimal reinsurance contracts only considers the interests of one party. This article combines the proportional and stop-loss reinsurance contracts and introduces a new reinsurance contract called proportional…
Optimal exit strategies of CPT gamblers in unfair gambles
problem Optimal exit strategies of gamblers with CPT preferences in games with strictly negative expected payoffs
method Formulating the problem as an optimal stopping problem on asymmetric random walks, applying geometric transformation, randomized strategies, and changing the decision variable
result The unfair problem in the infinite time horizon has finite values for a wide range of CPT parameter specifications
Optimizes investment under uncertain time horizons with non-concave utility.
problem Optimizing investment decisions with non-concave utility and uncertain time horizons.
method Established necessary and sufficient conditions for optimality, suggested recursive procedure for non-concave utility.
result Optimal investment strategies under uncertain time horizons exhibit multimodal distribution, indicating flexibility in switching between local maximizers.
Solves optimal stopping problem with Poisson constraints using jumps.
problem Optimal stopping with Poisson constraints and jumps.
method Penalized backward stochastic differential equation (PBSDE) with jumps, decomposition method based on Jacod-Pham, comparison theorem of BSDEs with jumps.
result Solves American option pricing in nonlinear markets with Poisson constraints.
Inference of stochastic time series with missing dataphysics.data-an
Method infers dynamics from incomplete time series data.
problem Challenges in inferring stochastic dynamics from time series with missing data.
method Expectation Maximization (EM) algorithm that iterates between E-step and M-step.
result The EM algorithm effectively recovers missing data points and infers underlying network models from real neuronal activities.
Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as…
Optimal timing for converting savings into annuities considering mortality risk.
problem Determining the best time to annuitize retirement savings under stochastic mortality.
method Formulated as a three-dimensional optimal stopping problem, reduced to nested one-dimensional problems, solved using PDMP structure.
result Rich structure for the optimal annuitization rule, covering various parameter specifications.
Quantum algorithm speeds up nested expectation estimation by nearly quadratically.
problem Estimating repeatedly nested expectations with quantum computing.
method Proposes a quantum algorithm achieving nearly quadratic speedup over classical methods.
result Achieves nearly quadratic speedup for RNEs, up to logarithmic factors.
This study analyzes AdaGrad's stability and convergence in non-convex optimization.
problem Lack of theoretical analysis for AdaGrad in non-convex optimization.
method Novel stopping time-based techniques from probability theory.
result Established stability and derived convergence rates for AdaGrad.
Equilibrium found for multi-agent trading with transaction costs.
problem Designing a trading equilibrium for multiple agents with transaction costs.
method Proving the existence of a continuous-time Radner equilibrium with incentives and transaction costs.
result Each agent optimally trades for a specific time interval before stopping, influenced by transaction costs.
A deep learning approach for computations of exposure profiles for high-dimensional Bermudan optionsq-fin.CP
Deep learning approximates Bermudan option exposures and future values.
problem Computing accurate expected and future exposures for high-dimensional Bermudan options.
method Neural network-based approach combining Deep Optimal Stopping and regression.
result Neural network approximations of pathwise option values are more accurate.
This paper extends stock trading results to include stop-loss orders.
problem Generalizing stock trading results with stop-loss orders.
method Geometric Brownian motion model, affine feedback controller, closed-form expression for cumulative distribution function.
result Affine feedback controller with stop-loss order generalizes results without stop-loss orders.
Optimal double stopping timemath.PR
We consider the optimal double stopping time problem defined for each stopping time by $v(S)=\esssup\{E[ψ(τ_1, τ_2) | \F_S], τ_1, τ_2 \geq S \}$. Following the optimal one stopping time problem, we study the existence of optimal stopping times and give a method to compute them. The key point is the construction of …
In this paper we study the problem of stopping a Brownian bridge in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 20…